We have compiled an extensive data set on potential parts of the Monoceros tidal stream and performed an exhaustive survey of dwarf galaxy semianalytic orbits in order to constrain its orbital properties. The best-fit orbits are subsequently realized as self-consistent N-body simulations in order to reproduce the spatial and velocity distribution of satellite debris. We find that all kinematic and geometric constraints can be fit by a single stream allowing for multiple wraps. The orbital eccentricity and inclination of the progenitor are strongly constrained to be e = 0.10 ± 0.05 and i = 25° ± 5°. Ten new estimates of proper motions from the Sloan Digital Sky Survey clearly exclude all retrograde orbits. Particles lost by the satellite populate two nearly concentric rings, naturally explaining the detection of stream stars at both 6-8 kpc (Ibata et al.; Newberg et al.) and 12-18 kpc (the Tri/And stream; Rocha-Pinto et al.) from the Sun. We have attempted to predict the present location of the Monoceros stream progenitor using different information: (1) the kinematical and spatial distribution of detections, and (2) the different mean metallicity in the inner and the outer rings. Because of the lack of observational data in the whole range of Galactic latitudes, the geometrical/kinematical constraints lead to a wide range of possible locations. By associating older parts of the model stream with lower metallicity parts of the observed data, we argue in favor of a current location of l ∼ 245°, b ∼ -18°, with a distance to the Sun rs ≃ 15 kpc. The mass of the progenitor has been poorly constrained because of the slow orbital decay. Similar fits have been obtained for masses (3-9) × 108 M⊙. We have analyzed the possible common origin of the Canis Major dwarf and the Monoceros stream. The Canis Major dwarf moves on a prograde, nearly circular orbit (e ≃ 0.16) in the Milky Way disk (i ≃ 4 -4 +14). deg). This orbital inclination is too low to account for the large vertical dispersion of stream stars. However, the bimodal distribution of radial velocities in the central region found by Martin et al. probably indicates that their selection criteria for identifying dwarf stars lead to a contamination of background stars. In that case, the kinematical data outlined above might result in an underestimate of the orbital inclination. Finally, the distance estimation to Canis Major dwarf is around a factor of 2 smaller than that obtained from our model. Unfortunately, the possible identification of the Monoceros stream progenitor in Canis Major remains unclear.